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Volume 12 (2016) PROGRESS IN PHYSICS Issue 1 (January) On an Apparent Resolution of the Catt Question Stephen J. Crothers Tasmania, Australia. E-mail: steve@plasmaresources.com Over a number of years there have been some attempts to answer the Catt Question within the context of classical electromagnetic theory. None of the authors of these attempts agree on the answer to the Catt Question, even though they all invoke the very same theory. An attempt at answering the Catt Question appeared in the journal Physics Education in 2013, penned by M. Pieraccini and S. Selleri, as a mathematical rendition of their earlier non-mathematical version published in IEEE Antennas and Propagation Magazine, 2012. The explanation by these two Authors contains violations of classical electromagnetic theory, although they claim to have satisfactorily answered the Catt Question by means of classical electromagnetic theory. The arguments adduced by Pieraccini and Selleri are therefore invalid. 1 Introduction In their article [1] “An apparent paradox: Catt’s anomaly”, the Italian authors Pieraccini and Selleri* refer to the Catt Ques- tion as “ Catt’s Anomaly’. Their earlier paper is titled ‘Catt’s Anomaly’ [2], Although until 2001 “The Catt Question” was called “The Catt Anomaly”, it was in fact always a question, to be answered. The Catt Question [3] pertains to the propagation of a Transverse Electromagnitic (TEM) wave along a transmis- sion line. Upon closure of a switch, the TEM wave (step) travels at the speed of light between the conducting wires of the transmission line, from battery to load, as depicted in Fig. 1. An electric field E appears between the conductors, di- rected from the top wire to the bottom wire. This electric field is orthogonal to the two parallel wires and moves towards the load; thus there are positive charges on the top conductor and negative charges on the bottom conductor in the region of the transverse electric field. The Catt Question is: Where does this new charge come from? [3], 2 Electron current According to classical electromagnetic theory and circuit the- ory, electric current in metallic wires is the flow of electrons in the wires (conductors), and a magnetic field is generated around the conducting wires according to the Right-Hand Rule. Since the TEM step travels at the speed of light to- wards the load, how does the current in the conducting wires keep pace with the TEM wave, if electrons cannot travel at the speed of light? The Authors [1] give the following answer, “The key idea of the explanation of this apparent paradox is related to the great number of elec- trons in metal. Although each single electron is "Massimiliano Pieraccini, Associate Professor, Department of Electron- ics and Telecommunications, University of Florence; Stefano Selleri, Assis- tant Professor, University of Florence. Fig. 1: An electric field points directly from the top conductor to the bottom conductor (from positive charge to negative charge). It is therefore orthogonal to the top and bottom parallel conductors. The transverse electric field travels from battery to load at the speed of light, subject to the dielectric medium between the wires. not able to travel at the speed of light, a great number of slow electrons are able to produce a current as fast as an electromagnetic wave trav- elling at the speed of light in the conductor.” What do they mean by “current”? They say here that elec- trons “ produce a current”. However, the Authors actually as- sume the classical electron flow along wires as the meaning of electric current in wires, and claim that this current travels along the conductors at the speed of light even though the drift speed of electrons in the wires is a snail’s pace (e.g. 2mm/s in 1.0mm copper wire [1]). Strangely, the flow of electrons, al- though very slow, produces an electron current that is “as fast as an electromagnetic wave travelling at the speed of light in the conductor” [1]: after all, a current of electrons is an electron current. This impossible duality occurs, they say, be- cause the free electron density in the conductors is very high, and they derive an equation for electron drift “velocity”. Electron drift velocity in a wire is proportional to the vec- tor electric field E„, in the wire, which supposedly causes the electron drift, v = ~pE w (0) and so the electron drift velocity and the electric field in the wire are collinear but point in opposite directions. The con- stant of proportionality p is called the mobility. 68 S. Crothers. On an Apparent Resolution of the Catt Question Issue 1 (January) PROGRESS IN PHYSICS Volume 12 (2016) The Authors begin with the following equation for elec- tron current, / = na 2 vqN , (1) where a is the radius of the conductors, v “ the drift velocity of the charges (in practice electrons, and the speed is much lower than the speed of light)” [1]*, q the elementary charge, and N the free electron density in the conductors. Since the current, they say, travels at the speed of light, in time At = Ax/c they obtain a passage of charge A Q along the top conductor, given by. make the transverse electric field the battery EMF to drive electrons along the wires; at equation (4). Then they introduce the “skin effect” [1]: “Up to this point, the current has been consid- ered constant in the wire section, but in reality the current flow tends to be bound to the portion of the conductor closer to the surface.” The equation for current in the wires they then give as, I = 2na6vqN, (lb) AQ = I At = I — , (2) c where Ax is the distance travelled by the TEM step in time At. This charge AQ the Authors call “an imbalance of charge” [1] because they say it is confined to a leading volume element of length Ax in the top conducting wire, and induces equal but opposite polarity charge on the bottom conducting wire. Using a cylindrical Gaussian surface they next apply Gauss’ Law to calculate the magnitude E of the electric field E due to AQ in the top conductor, A Q — =(2naAx)E, (3) 63 where eo is the permitivity of free space. Substituting AQ from equation (2) and / from equation (1) the Authors obtain, 2 cepE qNa (4) From equation (4) they conclude, “The notable point of this result is that the nec- essary speed decreases with the number of elec- trons per volume unit N. Therefore, a great num- ber of slow electrons are able to generate enough unbalanced charge to follow an electromagnetic wave travelling at much higher speed.” Thus electrons flow slowly in the conducting wires but the electron current in the wires is nevertheless flowing along the conductors at the speed of light. Although equation (4) follows from equations (1), (2) and (3) by purely mathematical operations, the transverse electric field E cannot drive electrons along the inside or outside of the wires. Equations (1), (2) and (4) imply flow of electrons along the wires, but the transverse electric field at equation (3) is orthogonal to the parallel axes of the top and bottom wires. According to classical electrodynamics, free electrons in a metallic conductor flow in the direction opposite to the direction of the electric field, according to equation (0), not orthogonal to the electric field (E + E„,). The Authors con- found battery EMF* with the transverse electric field, and so ’The Authors confound velocity with speed; the latter denoted by |v| = v ^What EMF is, is another question. where 6 is the skin depth, which is frequency dependent. With the “skin effect” they still argue that electrons flowing along the wire is electric current, orthogonal to the electric field they calculated at equation (3), and continue to make that trans- verse electric field the driver of the electrons in the conduct- ing wires. Using equations (lb), (2) and (3) they then obtain the electron drift speed, 2ceoE qN6 (5) although the 2 in the numerator should not in fact appear. 3 Conclusion Pieraccini and Selleri have not answered the Catt Question. On the one hand they treat current in the conducting wires as electron current but on the other hand they invoke the trans- verse electric field between the conducting wires to drive this electron current at the electron drift speed. Their analysis vio- lates the classical electromagnetic theory they use in their at- tempt to prove that what they call “Catt’s Anomaly” is merely an “apparent paradox” [1], The real paradox here is their claim that very slowly flowing electrons in the wires of a transmission line produce an electron current in those wires that travels at the speed of light, driven by an elecric field orthogonal to those wires. “If I have promised to deliver one dozen eggs to Oxford, one hour from now, Oxford being 100 miles away, there is no point in despatching ten dozen eggs in a vehicle which travels at only ten miles/h” [4], Submitted on December 13, 2015 / Accepted on December 14, 2015 References 1. Pieraccini M. and Selleri S. An apparent paradox: Catt’s anomaly. Physics Education, 2013, v.48 (6), 718-722. 2. Pieraccini M. and Selleri S. Catt’s anomaly. IEEE Antennas Propag. Mag., 2012, v. 54, no. 6, 242-244. 3. Catt I. The Catt Question, http://www.ivorcatt.co.uk/x54cl.htm 4. Catt I. The death of electric current. Wireless World, December 1982, 79-81. Accessed from http://www.ivorcatt.co.uk/xl8jl00.pdf S. Crothers. On an Apparent Resolution of the Catt Question 69